where \(_2F_1(a,b;c;z)\)  denotes Gauss' hypergeometric function [1]} and, from this (for comparison below),
\({\textsf {P}}[ X_{t-1}=0, X_{t+1}=0\mid X_t=2] &=[p^{\theta (1{-}\rho )}~{}_2F_1(\theta \rho , -x; \theta ; 1)]^2\\&= \left[p^{\theta (1{-}\rho )} (1{-}\rho )\right]^2\left[\frac{1+\theta (1{-}\rho )}{1+\theta }\right]^2.\) 
